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Perturbation of δ Potential on the Energy Level of a Onedimensional Quantum Well

Kaiyi Li

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

The paper studied the effect of a suddenly intruded δ potential on the energy level of a one-dimensional quantum well by perturbation method.It is found that δ potential has no effect on the odd parity states of the system but it makes the energy level of the even parity states have a constant shift,and the degree of the shift depends on the intensity of δ potential.

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quant-pharXiv:2602.20030•Feb 24, 2026
The paper studied the effect of a suddenly intruded δ potential on the energy level of a one-dimensional quantum well by perturbation method.It is found that δ potential has no effect on the odd parity states of the system but it makes the energy level of the even parity states have a constant shift,and the degree of the shift depends on the intensity of δ potential.
This chapter considers a particle with mass m and total energy E moving under the influence of the one-dimensional finite square well. An alternative method which turns out to be much more flexible when arrays of two or more wells are considered. The starting point of this alternative method is the boundary condition that the gradient of the wave function must be continuous at both edges of the well, and this implies that the gradient change across the well must be the same whether one uses the internal or external wave functions. Consider the array of three one-dimensional delta wells. This potential configuration can have up to three allowed states, depending on the well separation r. Since the potential is symmetrical about r = 0, the wave functions must have even or odd parity. Two of them have even parity while the third has odd parity.
Trial wave function was used to calculate ground state energy to obtain a relatively accurate value.After adding potential δ to one-dimensional harmonic oscillator,the odd parity stationary states of the harmonic oscillator still held and the even parity stationary states were no longer correct. Therefore,the perturbation theory and variational method were used respectively to calculate the approximate even parity stationary states,and variational method was used to solve Schrodinger equation.Results show that the calculated approximate even parity stationary states match well with the desired value,especially in excited state,the relative error is less than×10-3,and all the approximate values of energy for all levels are higher than desired value,which prove that it is simple and feasible by using perturbation theory or variational method to calculate approximate even parity stationary states with δ potential added.
We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of the solution of the harmonic oscillator problem.
An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.